The operator generated in <$E L sup 2 ( bold roman R )> by the differential expression <$E {roman sgn}~x~(D sup 2~+~aD~+~bI~+~c delta )> and boundary conditions at a point 0 is investigated. The class of boundary conditions for which such an operator is similar either to a normal or self-adjoint operator is described. In particular, a criterion of similarity of the operators <$E {roman sgn}~x~(D sup 2~+~aD~+~bI~+~c delta )> and <$E {roman sgn}~x~(D sup 2~+~bI~+~c delta prime )> to a normal operator is obtained. It turns out that these operators are similar to a normal one for almost all <$E c~symbol <174>~bold roman C>.

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